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1
07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Vagueness
• The Oxford Companion to Philosophy (1995):“Words like smart, tall, and fat are vague since in most contexts of use there is no bright line separating them from not smart, not tall, and not fat respectively …”
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Vagueness
• Imprecision vs. Uncertainty:The bottle is about half-full.vs.It is likely to a degree of 0.5 that the bottle is full.
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Sets
• Zadeh, L.A. (1965). Fuzzy SetsJournal of Information and Control
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Sets
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Definition
A fuzzy set is defined by a membership function that maps elements of a given domain (a crisp set) into values in [0, 1].
µµµµAAAA: U →→→→ [0, 1]µµµµAAAA ↔↔↔↔ A
0
1
20 40 Age30
0.5
young
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Representation
• Discrete domain::::high-dice score: 1:0, 2:0, 3:0.2, 4:0.5, 5:0.9, 6:1
• Continuous domain::::A(u) = 1 for u∈[0, 20]A(u) = (40 - u)/20 for u∈[20, 40]A(u) = 0 for u∈[40, 120]
0
1
20 40 Age30
0.5
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Representation
• α-cuts::::Aα = u | A(u) ≥ αAα+ = u | A(u) > α strong α-cut
A0.5 = [0, 30]
0
1
20 40 Age30
0.5
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Representation
• α-cuts::::Aα = u | A(u) ≥≥≥≥ αAα+ = u | A(u) >>>> α strong α-cut
A(u) = sup α | u ∈∈∈∈ Aα
A0.5 = [0, 30]
0
1
20 40 Age30
0.5
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Representation
• Support: : : : supp(A) = u | A(u) > 0 = A0+
• Core: : : : core(A) = u | A(u) = 1 = A1
• Height::::h(A) = supUA(u)
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Representation
• Normal fuzzy set:::: h(A) = 1
• Sub-normal fuzzy set:::: h(A) <<<< 1
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Membership Degrees
• Subjective definition
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Membership Degrees
• Voting model::::Each voter has a subset of U as his/her own crisp definition of the concept that A represents.A(u) is the proportion of voters whose crisp definitions include u. A defines a probability distribution on the power set of U across the voters.
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Membership Degrees
• Voting model:Voting model:Voting model:Voting model:
xxxxxxxxxx6xxxxxxxxx5
xxxxx4xx3
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P10P9P8P7P6P5P4P3P2P1
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Subset Relations
A ⊆ B iff A(u) ≤ B(u) for every u∈U
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Subset Relations
A ⊆ B iff A(u) ≤ B(u) for every u∈U
A is more specific than B
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Subset Relations
A ⊆ B iff A(u) ≤ B(u) for every u∈U
A is more specific than B
“X is A” entails “X is B”
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Operations
• Standard definitions::::Complement: A(u) = 1 −−−− A(u)Intersection: (A∩∩∩∩B)(u) = min[A(u), B(u)]Union: (A∪∪∪∪B)(u) = max[A(u), B(u)]
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Set Operations
• Example::::not young = youngnot old = oldmiddle-age = not young∩∩∩∩not oldold = ¬¬¬¬young
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Numbers
• A fuzzy number A is a fuzzy set on R:A must be a normal fuzzy set
Aα must be a closed interval for every α∈(0, 1]
supp(A) = A0+ must be bounded
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Basic Types of Fuzzy Numbers
1
0
1
0
1
0
1
0
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Basic Types of Fuzzy Numbers
1
0
1
0
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
• Extension principle for fuzzy sets::::f: U1×...×Un → V
induces
g: U1×...×Un → V ~~~
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
• Extension principle for fuzzy sets::::f: U1×...×Un → V
induces
g: U1×...×Un → V [g(A1,...,An)](v) = sup(u1,...,un) | v = f(u1,...,un)minA1(u1),...,An(un)
~~~
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
• EP-based operations: : : : (A + B)(z) = sup(x,y) | z = x+yminA(x),B(y)
(A −−−− B)(z) = sup(x,y) | z = x-yminA(x),B(y)(A * B)(z) = sup(x,y) | z = x*yminA(x),B(y)(A / B)(z) = sup(x,y) | z = x/yminA(x),B(y)
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
• EP-based operations::::
+∞
1
about 2 more or less 6 = about 2.about 3about 3
02 3 6
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
• Discrete domains:A = xi: A(xi) B = yi: B(yi)
A °°°° B = ?
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
• Interval-based operations: : : : (A ° B)α = Aα
° Bα
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
• Arithmetic operations on intervals::::[a, b]°[d, e] = f°g | a ≤≤≤≤ f ≤≤≤≤ b, d ≤≤≤≤ g ≤≤≤≤ e
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
• Arithmetic operations on intervals::::[a, b]°[d, e] = f°g | a ≤≤≤≤ f ≤≤≤≤ b, d ≤≤≤≤ g ≤≤≤≤ e[a, b] + [d, e] = [a + d, b + e][a, b] −−−− [d, e] = [a −−−− e, b −−−− d][a, b]*[d, e] = [min(ad, ae, bd, be), max(ad, ae, bd, be)][a, b]/[d, e] = [a, b]*[1/e, 1/d] 0∉∉∉∉[d, e]
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
+∞
1
about 2about 2 + about 3 = ?
about 2 × about 3 = ?
about 3
02 3
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Operations of Fuzzy Numbers
• Discrete domains:A = xi: A(xi) B = yi: B(yi)
A °°°° B = ?
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Possibility Theory
• Possibility vs. Probability
• Possibility and Necessity
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Possibility Theory
• Zadeh, L.A. (1978). Fuzzy Sets as a Basis for a Theory of PossibilityJournal of Fuzzy Sets and Systems
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Possibility Theory
• Membership degree = possibility degree
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Possibility Theory
• Axioms:
0 ≤≤≤≤ Pos(A) ≤≤≤≤ 1Pos(Ω) = 1 Pos(∅∅∅∅) = 0Pos(A ∪∪∪∪ B) = max[Pos(A), Pos(B)]Nec(A) = 1 – Pos(A)
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Possibility Theory
• Derived properties:
Nec(Ω) = 1 Nec(∅∅∅∅) = 0Nec(A ∩∩∩∩ B) = min[Nec(A), Nec(B)]max[Pos(A), Pos(A)] = 1min[Nec(A), Nec(A)] = 0Pos(A) + Pos(A) ≥≥≥≥ 1Nec(A) + Nec(A) ≤≤≤≤ 1Nec(A) ≤≤≤≤ Pos(A)
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Possibility Theory
r(u) = 1 for every u∈∈∈∈UTotal ignorance:::: p(u) = 1/|U| for every u∈∈∈∈U
Probability-possibility consistency principle:::: Pro(A) ≤≤≤≤ Pos(A)
Pos(A) + Pos(A) ≥≥≥≥ 1Pro(A) + Pro(A) = 1
Pos(A∪∪∪∪B) = max[Pos(A), Pos(B)]Additivity:::: Pro(A∪∪∪∪B) = Pro(A) + Pro(B) −−−− Pro(A∩∩∩∩B)
maxu∈∈∈∈Ur(u) = 1Normalization:::: ∑u∈∈∈∈Up(u) = 1
r: U → [0, 1]Pos(A) = maxu∈∈∈∈Ar(u)
p: U → [0, 1]Pro(A) = ∑u∈∈∈∈Ap(u)
PossibilityProbability
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Relations
• Crisp relation::::R(U1, ..., Un) ⊆⊆⊆⊆ U1× ... ×Un
R(u1, ..., un) = 1 iff (u1, ..., un) ∈∈∈∈ R or = 0 otherwise
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Relations
• Crisp relation::::R(U1, ..., Un) ⊆⊆⊆⊆ U1× ... ×Un
R(u1, ..., un) = 1 iff (u1, ..., un) ∈∈∈∈ R or = 0 otherwise
• Fuzzy relation is a fuzzy set on U1×××× ... ××××Un
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Relations
• Fuzzy relation::::U1 = New York, Paris, U2 = Beijing, New York, LondonR = very far
R = (NY, Beijing): 1, ...
.3.6London
.70NY
.91BeijingParisNY
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Multivalued Logic
• Truth values are in [0, 1]
• Lukasiewicz::::¬¬¬¬a = 1 −−−− aa ∧∧∧∧ b = min(a, b)a ∨∨∨∨ b = max(a, b)a ⇒⇒⇒⇒ b = min(1, 1 −−−− a + b)
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Logic
if x is A then y is Bx is A*------------------------y is B*
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Logic
• View a fuzzy rule as a fuzzy relationif x is A then y is B R(u, v) ≡≡≡≡ A(u) ⇒⇒⇒⇒ B(v)x is A* A*(u)
------------------------ ----------------------------------------y is B* B*(v)
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Logic
• Measure similarity of A and A*if x is A then y is Bx is A*
------------------------y is B*
B* = B + ∆(A/A*)
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Controller
• As special expert systems
• When difficult to construct mathematical models
• When acquired models are expensive to use
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Controller
IF the temperature is very high
AND the pressure is slightly low
THEN the heat change should be sligthly negative
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzy Controller
Controlledprocess
Defuzzificationmodel
Fuzzificationmodel
Fuzzy inference engine
Fuzzy rule base
actions
conditions
FUZZY CONTROLLER
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Fuzzification
1
0
x0
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Defuzzification
• Center of Area:
x = (∑A(z).z)/∑A(z)
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Defuzzification
• Center of Maxima:
M = z | A(z) = h(A)
x = (min M + max M)/2
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Defuzzification
• Mean of Maxima:
M = z | A(z) = h(A)
x = ∑z/|M|
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07 December 2009
Cao Hoang Tru
CSE Faculty - HCMUT
Exercises
• In Klir’s FSFL: 1.9, 1.10, 2.11, 4.5, 5.1 (a)-(b), 8.6, 12.1.
